On the structure of almost Yamabe solitons

TL;DR

This work extends the study of Yamabe solitons to almost Yamabe solitons without the gradient constraint, analyzing when the soliton field $X$ must be Killing or parallel under compact and noncompact completeness assumptions. It develops a robust conformal-structure framework, using identities for $\nabla_Y X=(R-\rho)Y+\Phi(Y)$ and associated Bochner-type and divergence arguments, to derive sharp rigidity results. A key contribution is establishing local warped product or twisted product structures when $X$ is closed, thereby generalizing existing gradient-based structure results to the non-gradient setting. These findings clarify how the interaction between the scalar curvature $R$, the soliton function $\rho$, and the conformal factor controls the global geometry of almost Yamabe solitons, with implications for the classification of such solitons on compact and noncompact manifolds.

Abstract

In this paper, we study structures of almost Yamabe solitons which are not necessarily gradient. First, we investigate conditions that both compact and noncompact almost Yamabe solitons become trivial solitons which means the given vector field is a Killing vector field. Second, we show that an almost Yamabe soliton whose vector field is closed admits a local warped product structure with a one-dimensional base. This result can be considered as a generalization of a result in \cite{c-s-z} and \cite{c-m-m}

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Theorem 3.2